A new form of the Boussinesq equations with improved linear dispersion characteristics. Part 2. A slowly-varying bathymetry
TL;DR: In this paper, a new form of the Boussinesq equations applicable to irregular wave propagation on a slowly varying bathymetry from deep to shallow water is introduced, which incorporate excellent linear dispersion characteristics, and are formulated and solved in two horizontal dimensions.
About: This article is published in Coastal Engineering.The article was published on 1992-12-01. It has received 783 citations till now. The article focuses on the topics: Boussinesq approximation (water waves) & Shallow water equations.
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TL;DR: In this article, a third-generation numerical wave model to compute random, short-crested waves in coastal regions with shallow water and ambient currents (Simulating Waves Nearshore (SWAN)) has been developed, implemented, and validated.
Abstract: A third-generation numerical wave model to compute random, short-crested waves in coastal regions with shallow water and ambient currents (Simulating Waves Nearshore (SWAN)) has been developed, implemented, and validated. The model is based on a Eulerian formulation of the discrete spectral balance of action density that accounts for refractive propagation over arbitrary bathymetry and current fields. It is driven by boundary conditions and local winds. As in other third-generation wave models, the processes of wind generation, whitecapping, quadruplet wave-wave interactions, and bottom dissipation are represented explicitly. In SWAN, triad wave-wave interactions and depth-induced wave breaking are added. In contrast to other third-generation wave models, the numerical propagation scheme is implicit, which implies that the computations are more economic in shallow water. The model results agree well with analytical solutions, laboratory observations, and (generalized) field observations.
3,625 citations
Cites background from "A new form of the Boussinesq equati..."
...…are usually based on Hamiltonian equations [e.g., Miles, 1981; Radder, 1992], Boussinesq equations [e.g., Peregrine, 1966; Freilich and Guza, 1984; Madsen and Sørensen, 1992], or on the mild-slope equation (e.g., Berkhoff [1972] or its parabolic version, e.g., Radder [1979] and Kirby [1986])....
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TL;DR: In this paper, a new form of the Boussinesq equations is derived using the velocity at an arbitrary distance from the still water level as the velocity variable instead of the commonly used depth-averaged velocity.
Abstract: Boussinesq‐type equations can be used to model the nonlinear transformation of surface waves in shallow water due to the effects of shoaling, refraction, diffraction, and reflection. Different linear dispersion relations can be obtained by expressing the equations in different velocity variables. In this paper, a new form of the Boussinesq equations is derived using the velocity at an arbitrary distance from the still water level as the velocity variable instead of the commonly used depth‐averaged velocity. This significantly improves the linear dispersion properties of the Boussinesq equations, making them applicable to a wider range of water depths. A finite difference method is used to solve the equations. Numerical and experimental results are compared for the propagation of regular and irregular waves on a constant slope beach. The results demonstrate that the new form of the equations can reasonably simulate several nonlinear effects that occur in the shoaling of surface waves from deep to shallow w...
1,112 citations
Book•
01 Feb 2010TL;DR: The SWAN wave model as discussed by the authors is a wave model based on linear wave theory (SWAN) for oceanic and coastal waters, and it has been shown to be effective in detecting ocean waves.
Abstract: 1. Introduction 2. Observation techniques 3. Description of ocean waves 4. Statistics 5. Linear wave theory (oceanic waters) 6. Waves in oceanic waters 7. Linear wave theory (coastal waters) 8. Waves in coastal waters 9. The SWAN wave model Appendices References Index.
874 citations
TL;DR: In this article, an extended Boussinesq model for surf zone hydrodynamics in two horizontal dimensions is implemented and verified using an eddy viscosity term.
Abstract: In this paper, we focus on the implementation and verification of an extended Boussinesq model for surf zone hydrodynamics in two horizontal dimensions The time-domain numerical model is based on the fully nonlinear Boussinesq equations As described in Part I of this two-part paper, the energy dissipation due to wave breaking is modeled by introducing an eddy viscosity term into the momentum equations, with the viscosity strongly localized on the front face of the breaking waves Wave runup on the beach is simulated using a permeable-seabed technique We apply the model to simulate two laboratory experiments in large wave basins They are wave transformation and breaking over a submerged circular shoal and solitary wave runup on a conical island Satisfactory agreement is found between the numerical results and the laboratory measurements
659 citations
Cites background from "A new form of the Boussinesq equati..."
...Recent advances in both computer technology and dispersive, nonlinear long-wave theory ( Madsen and Sorensen 1992; Nwogu 1993; Wei et al. 1995; Madsen and Scha ¨ffer 1998; Chen et al. 1998) now permit the use of Boussinesq wave models for large nearshore regions and allow the averaging of model results to predict wave-induced mean flows if wave breaking is incorporated into the model....
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TL;DR: A high-order adaptive time-stepping TVD solver for the fully nonlinear Boussinesq model of Chen (2006), extended to include moving reference level as in Kennedy et al. (2001).
486 citations
References
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Book•
01 Jan 1983
TL;DR: In this article, the authors present selected theoretical topics on ocean wave dynamics, including basic principles and applications in coastal and offshore engineering, all from a deterministic point of view, and the bulk of the material deals with the linearized theory.
Abstract: The aim of this book is to present selected theoretical topics on ocean wave dynamics, including basic principles and applications in coastal and offshore engineering, all from the deterministic point of view. The bulk of the material deals with the linearized theory.
2,003 citations
TL;DR: In this paper, the Boussinesq equations for long waves in water of varying depth are derived for small amplitude waves, but do include non-linear terms, and solutions have been calculated numerically for a solitary wave on a beach of uniform slope, which is also derived analytically by using the linearized long-wave equations.
Abstract: Equations of motion are derived for long waves in water of varying depth. The equations are for small amplitude waves, but do include non-linear terms. They correspond to the Boussinesq equations for water of constant depth. Solutions have been calculated numerically for a solitary wave on a beach of uniform slope. These solutions include a reflected wave, which is also derived analytically by using the linearized long-wave equations.
1,352 citations
TL;DR: In this paper, a new form of the Boussinesq equations is introduced in order to improve their dispersion characteristics, and a numerical method for solving the new set of equations in two horizontal dimensions is presented.
694 citations
TL;DR: Boussinesq type equations with improved linear dispersion characteristics are derived and applied to study wave-wave interaction in shallow water in this article, where weakly nonlinear solutions are formulated in terms of Fourier series with constant or spatially varying coefficients for two purposes: to derive higher order boundary conditions for regular and irregular wave trains and to derive evolution equations on constant or variable water depth.
168 citations
01 Dec 1982
TL;DR: In this article, a model of water waves that describes wave propagation over long distances accurately, at low cost, and for a wide variety of physical situations are given, using exact prognostic equations, and a high-order expansion to relate variables at each time step.
Abstract: Details of a new model of water waves that describes wave propagation over long distances accurately, at low cost, and for a wide variety of physical situations are given. The analysis and numerical methods selected for computer solution are given in some detail. The model uses exact prognostic equations, and a high-order expansion to relate variables at each time step. The accuracy of the model is demonstrated most completely for solitary wave propagation, where model results are compared to exact results. It is found that the model results are much more accurate for high solitary waves than are earlier, Boussinesq-type theories, and give good results for waves so high that they are almost breaking. The capability of the model to treat a variety of situations is demonstrated for colliding solitary waves, nonlinear dispersive wave trains, waves in channels of varying breadth, and undular bores. Formally, the model incorporates nonlinear long wave theory exactly, incorporates enough dispersion to describe linear waves with fourth-order precision, so that both shallow water waves and deep water waves are included, and describes accurately waves for which dispersive and nonlinear effects are both important.
164 citations